A Mixed 0-1 Linear Programming Approach to the Computation of All Pure-Strategy Nash Equilibria of a Finite n-Person Game in Normal Form
A main concern in applications of game theory is how to effectively select a Nash equilibrium, especially a pure-strategy Nash equilibrium for a finite -person game in normal form. This selection process often requires the computation of all Nash equilibria. It is well known that determining whether a finite game has a pure-strategy Nash equilibrium is an NP-hard problem and it is difficult to solve by naive enumeration algorithms. By exploiting the properties of pure strategy and multilinear terms in the payoff functions, this paper formulates a new mixed 0-1 linear program for computing all pure-strategy Nash equilibria. To our knowledge, it is the first method to formulate a mixed 0-1 linear programming for pure-strategy Nash equilibria and it may work well for similar problems. Numerical results show that the approach is effective and this method can be easily distributed in a distributed way.
Nash equilibrium, as defined in , plays a fundamental role in the development of game theory and its diverse applications in areas such as biology, computer science, economics, and management and social sciences. In these applications, a main concern is how to effectively select a Nash equilibrium, especially a pure-strategy Nash equilibrium for a finite -person game in normal form. To tackle this problem, many contributions have been made in the literature.
In recent years, most related work is focused on finding mixed-strategy Nash equilibrium. In terms of computational complexity, it was shown in [1, 2] that the problem of finding one mixed-strategy Nash equilibrium is PPAD-complete. The first method for computing Nash equilibrium of a two-person game was developed in  and was extended to computing Nash equilibria of -person games independently in [4, 5]. The existence of Nash equilibrium was established from an application of Brouwer and Kakutani fixed-point theorems in [6, 7]. To approximate fixed points of continuous mappings, simplicial methods were originated in  and substantially developed in the literature such as [9–15]. Simplicial methods were adapted for computing Nash equilibria in . Several more advanced simplicial methods for computing Nash equilibria were presented in [17–19]. A pivoting procedure was proposed in [20, 21] for computing the Nash equilibria of two-person games selected by the linear tracing procedure proposed in . This procedure was extended to -person games in . A survey of the literature on the computation of Nash equilibria before 1996 can be found in  and on computing equilibria for two-person games in . By exploiting differentiability of the problem, several methods have been developed for computing Nash equilibria in the literature. A smooth logarithmic tracing procedure with two parameters was proposed in . It was proved in  that the procedure converges to a Nash equilibrium as two parameters approach zero one after another. The linear tracing procedure together with well-chosen nonlinear transformations of variables of the system of the optimality conditions leads to a differentiable homotopy method in . As a result of Kohlberg and Mertens’ structure theorem , the global Newton method in  was combined with the homotopy method in  to obtain an algorithm for computing Nash equilibrium. A survey on homotopy methods for computing Nash equilibrium is referred to in . Some new related work has been developed in the literature [32–36].
All the algorithms mentioned above can only find a sample Nash equilibrium. Unfortunately, in practice, a game can have many Nash equilibria and which equilibrium should be effectively selected to play is a very challenging issue. The equilibrium selection process often requires the computation of all Nash equilibria of a game. To address this issue, several algorithms have been developed in the literature. A path-following algorithm was described in  to compute all Nash equilibria of a bimatrix game. A homotopy algorithm was presented in  to find all Nash equilibria of an -person game. Both of these algorithms are based on systems of polynomial equations. A survey on polyhedral homotopy continuation methods for all Nash equilibria can be found in . In , a mixed-integer program (MIP) was formulated to find Nash equilibria of a bimatrix game, which can be employed to compute all Nash equilibria of a bimatrix game. Recently, by enumerating vertices of polyhedrons, two algorithms were introduced in  for computing all Nash equilibria of a bimatrix game.
As a special example of the mixed-strategy Nash equilibrium, the pure-strategy Nash equilibrium has gotten more attention recently. The work of  shows that determining whether a game has a pure-strategy Nash equilibrium is NP-hard. The work of  presents a novel generic mapping between graphical games and Markov random fields so that pure Nash equilibria in the former can be found by statistical inference on the latter. Based on a new description format of game-stimulate-response pair, the work of  puts forward a constraints’ satisfaction-based algorithm on this data structure to compute pure Nash equilibrium of graphical game. The work of  develops an efficient algorithm, which is called Valued Nash Propagation, for computing pure strategy Nash equilibria that satisfy various criteria (such as the utilitarian or Nash-Bernoulli social welfare functions) in games with sparse interaction structure. Some other related work can be found in [2, 45, 46].
As a special case of integer programming, the mixed 0-1 linear programming, in which unknowns are binary, has many applications in operation research. Many classic problems, such as the knapsack problem, facility location problem, production planning problem, and time tabling problem, can be formulated as a mixed 0-1 linear programming. The research of these problems which has a long history and many commercial softwares, most of which are based on branch and bound method, cutting plane method, or branch and cut method, could solve this mixed 0-1 linear programming well.
This paper studies the computation of all pure-strategy Nash equilibria of a finite -person game in normal form. To solve this problem, we formulate a mixed 0-1 linear programming by exploiting the properties of pure strategy and multilinear forms in the payoff functions. An enumeration of all the feasible solutions of the mixed 0-1 linear program yields all pure-strategy Nash equilibria. Numerical results show that the method is promising. To our knowledge, it is the first method to formulate a mixed 0-1 linear programming for pure-strategy Nash equilibria and this method may work well for other similar problems. Furthermore, as an advantage of the formulation, one can apply distributed computation to dramatically speed up the computation.
The rest of the paper is organized as follows. Section 2 formulates a mixed 0-1 linear program for computing all pure-strategy equilibria. Section 3 presents numerical results to show that the approach is effective. The paper is concluded with some remarks in Section 4.
2. A Mixed 0-1 Linear Programming Formulation
In this section a mixed 0-1 linear programming will be formulated to all pure-strategy Nash equilibria of a finite game.
Let be the set of players. The pure strategy set of player is denoted by with . Given with , the set of all pure strategy profiles is . We denote the payoff function of player by . For , let . Then, can be rewritten as with . A mixed strategy of player is a probability distribution on denoted by . Let be the set of all mixed strategies of player . Then, . Thus, for , the probability assigned to pure strategy is equal to . Given with , the set of all mixed strategy profiles is . For , let . Then, can be rewritten as with . If is played, then the probability that a pure strategy profile occurs is . Therefore, for , the expected payoff of player is given by . With these notations, a finite -person game in normal form can be represented as or .
Definition 1 (Nash, ). A mixed strategy profile is a Nash equilibrium of game if for all and .
With this definition, an application of the optimality condition leads to the fact that is a Nash equilibrium if and only if there are and together with satisfying the system of where .
Lemma 2. Let be a given positive number such that Then, (1) is equivalent to
Proof. Let be a solution of the system (3). Suppose that in (3). Then, and . Thus, . Suppose that = 1 in (3). Then, . Thus, . Therefore, is a solution of the system (1).
Let be a solution of the system (1). Since , there must exist some such that . From , we get that . Thus, . Therefore, for any , we obtain from the system (1) that
Suppose that . Let . Thus, Suppose that . Let . Thus, Therefore, is a solution of the system (3). This completes the proof.
This lemma implies that finding a pure-strategy Nash equilibrium is equivalent to finding a solution to the system of For any , one can obtain from that Let for . Then, Substituting this into (7) yields Observe that since . This property leads to the following result.
Lemma 3. For and , the system of is equivalent to the system of
Proof. Let be a solution of the system (12). Suppose that , for all . Then,
Suppose that , . Then, Thus, . Therefore, is a solution of the system (11).
Let be a solution of the system (11). Suppose that , for all . Then, . Thus, Suppose that , . Then, . Thus, Therefore, is a solution of the system (12). This completes the proof.
The above idea is illustrated in the following example.
Example 5. Consider a three-player game , where , , , and are given by
Let . The mixed 0-1 linear program for this game is given by
3. Numerical Results
In this section, some numerical results are presented. Under the software Microsoft Visual Studio 2008 and ILOG CPLEX, we use C++ to call ILOG CPLEX API functions which are based on the branch and cut method to solve the mixed 0-1 linear program (17). The MIP (mixed integer programming) search method, which is a dynamic search strategy or branch and cut strategy in ILOG CPLEX, is determined by the ILOG CPLEX automatically. All other parameters are also set automatically by ILOG CPLEX itself.
We have run our code on a workstation of Lenovo ThinkStation D20 4155-BM4 with 16 processors and 16 G RAM. In the presentation of numerical results, some symbols are explained as follows: Num: the number of players; Num: the number of strategies for each player; NumEquilibra: the number of pure-strategy Nash equilibria for the instance; Time: the total computational time to solve the problem.
Example 1. Consider a three-player game , where . The number of strategies for each player, , is generated from 3 to 45 randomly. The are generated randomly too. There are three different ranges for in this example, which are from 0 to 10, from 0 to 50, and from 0 to 100.
Example 2. Consider a five-player game , where . The number of strategies for each player, , is generated from 3 to 7 randomly. The are generated randomly too. There are three different ranges for in this example, which are from 0 to 10, from 0 to 50, and from 0 to 100.
From Tables 1, 2, and 3, one can see that this mixed 0-1 linear programming method can find all pure-strategy Nash equilibria of three-person game effectively. For example, in Table 1 the problem 30 with 45 strategies can be solved in 16992.10 seconds and the number of equilibria is 26. It has been shown that every -person game can be reduced to a three-person game in polynomial time in . However, some five-person games are still solved by this method in our examples in order to test the effectiveness of this mixed 0-1 linear programming approach. The numerical results for five-person can be obtained from Tables 4, 5, and 6. All the numerical results show that this new method can handle all pure-strategy Nash equilibria of -person successfully. However, the computation time increases quickly with the increasing of the problems’ dimension. To tackle this problem, the distributed computation, which is a salient feature of the integer programming, will be considered in our next work. Besides, some similar problems with multilinear terms could be solved by this method too.
In this paper, a mixed 0-1 linear programming approach has been proposed to find all pure-strategy Nash equilibria of a finite -person game in normal form. This method is based on the properties of pure strategy and multilinear terms in the payoff functions. Some numerical results have been given too. The results are promising. However, there is still some interesting future work to be done in the next step. For one thing, as a nice feature of integer programming, the computation will be implemented in a distributed way, which can significantly speed up the efficiency. For another, more research about the properties of strategy and multilinear terms has to be done in order to compute more general Nash equilibria, mixed-strategy Nash equilibria.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by the RCN (Research Council of Norway) DECOFF project number 225231/O70 and by GRF: CityU 112910 of Hong Kong SAR Government.
C. B. Garcia and W. I. Zangwill, Pathways to Solutions, Fixed Points, and Equilibria, Prentice Hall, Englewood Cliffs, NJ, USA, 1981.
M. J. Todd, The Computation of Fixed Points and Applications, vol. 1 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 1976.View at: MathSciNet
C. B. Garcia, C. E. Lemke, and H. Luethi, “Simplicial approximation of an equilibrium point of noncooperative -person games,” Mathematical Programming, vol. 4, pp. 227–260, 1973.View at: Google Scholar
G. van der Laan, A. J. J. Talman, and L. van der Heyden, “Simplicial variable dimension algorithms for solving the nonlinear complementarity problem on a product of unit simplices using a general labelling,” Mathematics of Operations Research, vol. 12, no. 3, pp. 377–397, 1987.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. C. Harsanyi and R. Selten, A General Theory of Equilibrium Selection in Games, vol. 1, MIT Press, Cambridge, Mass, USA, 1988.View at: MathSciNet
S. H. Schanuel, L. K. Simon, and W. R. Zame, “The algebraic geometry of games and the tracing procedure,” Game Equilibrium Models, vol. 2, pp. 9–43, 1991.View at: Google Scholar
S. Yin, H. Luo, and S. Ding, “Real-time implementation of fault-tolerant control systems with performance optimization,” IEEE Transactions on Industrial Electronics, vol. 61, no. 5, pp. 2402–2411, 2013.View at: Google Scholar
T. Sandholm, A. Gilpin, and V. Conitzer, “Mixed-integer programming methods for finding Nash equilibria,” in Proceedings of the 20th National Conference on Artificial Intelligence, pp. 495–501, July 2005.View at: Google Scholar
G. Gottlob, G. Greco, and F. Scarcello, “Pure Nash equilibria: hard and easy games,” in Proceedings of the 9th Conference on Theoretical Aspects of Rationality and Knowledge, pp. 215–230, ACM, 2003.View at: Google Scholar
M. Jiang, “Finding pure Nash equilibrium of graphical game via constraints satisfaction approach,” in Combinatorics, Algorithms, Probabilistic and Experimental Methodologies, pp. 483–494, Springer, New York, NY, USA, 2007.View at: Google Scholar
A. C. Chapman, A. Farinelli, E. M. de Cote, A. Rogers, and N. R. Jennings, “A distributed algorithm for optimising over pure strategy Nash equilibria,” in Proceedings of the 24th AAAI Conference on Artificial Intelligence, pp. 749–755, Atlanta, Ga, USA, July 2010.View at: Google Scholar
G. Gottlob, G. Greco, and T. Mancini, “Complexity of pure equilibria in bayesian games,” in Proceedings of the International Joint Conference on Artificial Intelligence, pp. 1294–1299, 2007.View at: Google Scholar
A. C. Chapman, Control of large distributed systems using games with pure strategy Nash equilibria [Ph.D. thesis], University of Southampton, 2009.